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A Domain Wall Solution by Perturbation of the Kasner Spacetime

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A Domain Wall Solution by Perturbation of the Kasner Spacetime A Domain Wall Solution by Perturbation of the Kasner Spacetime George Kotsopoulos 1906-373 Front St. West, Toronto, ON M5V 3R7 [email protected] and Charles C. Dyer Dept. of Physical and Environmental Sciences, University of Toronto Scarborough, and Dept. of Astronomy and Astrophysics, University of Toronto, Toronto, Canada [email protected] January 24, 2012 Received ; accepted arXiv:1201.5362v1 [gr-qc] 25 Jan 2012 –2– Abstract Plane symmetric perturbations are applied to an axially symmetric Kasner spacetime which leads to no momentum flow orthogonal to the planes of symme- try. This flow appears laminar and the structure can be interpreted as a domain wall. We further extend consideration to the class of Bianchi Type I spacetimes and obtain corresponding results. Subject headings: general relativity, Kasner metric, Bianchi Type I, domain wall, perturbation –3– 1. Finding New Solutions There are three principal exact solutions to the Einstein Field Equations (EFE) that are most relevant for the description of astrophysical phenomena. They are the Schwarzschild (internal and external), Kerr and Friedman-LeMaître-Robertson-Walker (FLRW) models. These solutions are all highly idealized and involve the introduction of simplifying assumptions such as symmetry. The technique we use which has led to a solution of the EFE involves the perturbation of an already symmetric solution (Wilson and Dyer 2007). Whereas standard perturbation techniques involve specifying the forms of the energy-momentum tensor to determine the form of the metric, we begin by defining a new metric with an applied perturbation: gab =g ˜ab + hab (1) where g˜ab is a known symmetric metric and hab is the applied perturbation. We then use this new metric to find the components of the Einstein tensor, Gab, in terms of the metric components. Next, we invert the EFE, where now Tab = Gab/κ, to ascertain whether or not components of the energy-momentum tensor, Tab, exist to satisfy the perturbation. If such an energy-momentum tensor is physically acceptable, we then try to limit the behaviour of our perturbation by imposing conditions to model particular astronomical phenomenon. For example, if we were modelling a galaxy-like structure, we could choose our density to scale as ρ r−2 from the galactic centre (Wilson and Dyer 2007). This makes the ∼ technique flexible and relevant in that we can model a physically viable structure. With these real physical constraints applied, we have a new metric. We now re-compute the energy-momentum components of this new metric and investigate any implications that may arise from our new exact solution to the EFE. –4– 2. The Base Solution The metric we choose to perturb is the Kasner spacetime (Kasner 1921). This metric describes an anisotropic universe that is a vacuum solution to the EFE classified as a Bianchi Type I universe. It is characterized by a spacetime that is anisotropically expanding (or contracting) in two directions while contracting (or expanding) in the third with space-like slices that are spatially flat and with a singularity occurring at t = 0. Hence, the Kasner spacetime is of special interest in cosmology since the standard cosmological solutions to the EFE near the cosmological singularity, such as the aforementioned FLRW model, can be described as a succession of Kasner epochs (Lifshitz and Khalatnikov 1963). Furthermore, by breaking a homogeneity along one direction, Bianchi Type solutions can easily be generalized to spacetimes with G2 isotropy groups (Harvey 1990). The Kasner spacetime in Cartesian coordinates with metric signature (+, , ..., ) − − takes the form (in spacetime dimension D): D−1 2 2 2 2 ds = dt t Pj [dx ] (2) − j j=1 X where Pj are the Kasner exponents which satisfy: D−1 D−1 2 Pj =1 and Pj =1 (3) j=1 j=1 X X The first condition in (3) describes a plane whereas the second condition describes a sphere of dimension D 1. Thus, the Kasner exponents lie on a sphere of dimension D 2. For − − D = 4, at each t = constant hypersurface, there exists a flat 3-dimensional space, whose worldlines of constant x, y and z are time-like geodesics along which galaxies, or other test particles, can be imagined to move. Since the solution represents an anisotropically expanding (or contracting) universe, a volume element dV increases (or decreases) in time as g d3x = td3x, where g = det g . | | | ab| p –5– The Kasner spacetime in Cartesian coordinates in a 4 D spacetime takes the form: − ds2 = dt2 t2P1 dx2 t2P2 dy2 t2P3 dz2 (4) − − − with the same restrictions imposed by the Kasner exponents as given in (3). The form of the Kasner spacetime that we consider is the axisymmetric case with G2 isometry that occurs when two of the exponents are equal, which by (3), requires that P1 = P2 =2/3 and P3 = 1/3. With this restriction, the metric in cylindrical coordinates − becomes: ds2 = dt2 t4/3(dr2 + r2dφ2) t−2/3dz2 (5) − − 3. The Perturbations The set of perturbations we implement are only on the z-component of the Kasner spacetime represented in cylindrical coordinates (5). Clearly, the results we obtain in cylindrical coordinates will be similar to those in Cartesian coordinates since r2 = x2 + y2. For simplicity, we define the Kasner spacetime in cylindrical coordinates as: ds2 = dt2 t4/3(dr2 + r2dφ2) g dz2 (6) − − zz where gzz =g ˜zz + hab is our perturbed metric. We consider the three Cases shown in Table 1. Cases gzz =g ˜zz + hab I [t−2/3 + H] II [t−1/3 + H]2 III [t−2/3 + Z(z)T (t)] Table 1: Implemented perturbations. –6– In Cases I and II, we first investigate H as a differentiable function of z. We then investigate Cases I and II again, but where H is now made a differentiable function of t, r and z. In Case III, we investigate the result of a perturbation with a product separable differentiable solution. To aid in these computations, we use the REDUCE Computer Algebra System (Hearn 2009) with the REDTEN Tensor Analysis Package (Harper and Dyer 1994). 4. Results of the Perturbations Both Cases I and II produce a Gab with non-zero components. When the perturbation is H(z), the result produces non-zero terms for Gtt, Grr and Gφφ. All other terms, including Gzz, are zero. When the perturbation is H(t, r, z), a similar result is obtained for all cases, but with the addition of cross-components, G for a = b. These cross terms can ab 6 be made zero if we choose the perturbation to contain only first-order terms in H or by making ∂H/∂r = ∂H/∂φ = 0. Also, in cylindrical coordinates for all the cases reviewed, 2 Grr = r Gφφ, which is expected. The product separable perturbation of Case III reveals the same results as Case I and II; we arrive at non-zero terms for Gtt, Grr and Gφφ and a Gzz =0. Thus all perturbations that involved only the gzz term of the Kasner metric reveal that we will always obtain Gzz = 0 when considering axisymmetric or plane symmetric Kasner spacetimes, when P 1= P 2. Relating these results to the energy-momentum tensor Tab, both the Gzz and Gza = Gaz (for a = 0, 1, 2 ) components of the Einstein tensor being zero reveals that there is no { } momentum-flux across the z = constant surface regardless of the type of perturbation applied. Each of the applied perturbations resulted in the metric collapsing (or expanding) in the z-direction while expanding (or collapsing) in the x- and y-directions, as is expected –7– of the Kasner metric. But, as a result of the Gzz term being zero, there is no interaction between the stratified layers of the matter above or below the xy-plane. This is suggestive of laminar flow. Inverting the EFE to get Tab = Gab/κ, leads to Tab being diagonal but with no Tzz component. The energy-momentum tensor for an infinite, static plane-symmetric domain wall as suggested by Campanelli et al is: T = δ(z)diag(ρ, p, p, 0) (7) ab − − where ρ is the energy-density and p is the pressure. This energy-momentum ten- sor (Campanelli et al. 2003) corresponds to an infinite, static plane-symmetric domain wall (Kibble 1976) lying in the xy-plane. Domain walls correspond to a particular class of topological defects whereby the energy-density is trapped, and from a cosmological point of view, these over-dense regions could lead to structure formation (Brandenberger 1997). Furthermore, it has been suggested (Friedland et al. 2003) that the Universe may be dominated by a network of domain walls and these domain wall structures could represent an alternative view of dark energy theories. 5. Energy Conditions In order to rule out any non-physical solutions to the EFE, energy conditions are applied to the state of matter content for gravitational and non-gravitational fields. These energy conditions consist of the Weak, Null, Strong and Dominant energy conditions, which are coordinate-invariant constraints on the energy-momentum tensor. For the cases in which our Kasner spacetime was perturbed with H(z), we apply the least stringent of these conditions, the Null Energy Condition (NEC). This condition states –8– that for all future-pointing null vectors, ka: a b Tabk k ≧ 0 (8) This restriction implies that ρ + p ≧ 0, whereby the energy-density may be negative as long as there is a compensating pressure. If this condition is violated in any of the perturbed Kasner spacetimes, it would then indicate that our solutions are unstable. We choose the null vector field ka, in the Kasner spacetime to be, 2 1 n g 2 ka = − zz , 0, 0, n (9) s gtt ! where n ǫ R.
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